# A Theoretical Framework for an Efficient Normalizing Flow-Based Solution to the Electronic Schrodinger Equation

arXiv: https://arxiv.org/abs/2406.00047 · HTML: https://mlsanity.com/paper/2406.00047/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 57 references that are in the map.
- First posted on arXiv 28 May 2024.

## Builds on (most cited first, 26 of 26)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [Ab-Initio Solution of the Many-Electron Schr"odinger Equation with Deep Neural Networks](https://mlsanity.com/paper/1909.02487/) (2019)
- [Deep neural network solution of the electronic Schr"odinger equation](https://mlsanity.com/paper/1909.08423/) (2019)
- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [Sampling using SU(N) gauge equivariant flows](https://mlsanity.com/paper/2008.05456/) (2020)
- [Quantum Entanglement in Deep Learning Architectures](https://mlsanity.com/paper/1803.09780/) (2018)
- [Better, Faster Fermionic Neural Networks](https://mlsanity.com/paper/2011.07125/) (2020)
- [Generalizing Neural Wave Functions](https://mlsanity.com/paper/2302.04168/) (2023)
- [Gold-standard solutions to the Schr"odinger equation using deep learning: How much physics do we need?](https://mlsanity.com/paper/2205.09438/) (2022)
- [Discovering Quantum Phase Transitions with Fermionic Neural Networks](https://mlsanity.com/paper/2202.05183/) (2022)
- [Aspects of scaling and scalability for flow-based sampling of lattice QCD](https://mlsanity.com/paper/2211.07541/) (2022)
- [Normalizing flows for lattice gauge theory in arbitrary space-time dimension](https://mlsanity.com/paper/2305.02402/) (2023)
- [Neural tensor contractions and the expressive power of deep neural quantum states](https://mlsanity.com/paper/2103.10293/) (2021)
- [Electronic excited states in deep variational Monte Carlo](https://mlsanity.com/paper/2203.09472/) (2022)
- [Ab-initio study of interacting fermions at finite temperature with neural canonical transformation](https://mlsanity.com/paper/2105.08644/) (2021)
- [m^ast of two-dimensional electron gas: a neural canonical transformation study](https://mlsanity.com/paper/2201.03156/) (2022)
- [Transferable Neural Wavefunctions for Solids](https://mlsanity.com/paper/2405.07599/) (2024)
- [Convergence to the fixed-node limit in deep variational Monte Carlo](https://mlsanity.com/paper/2010.05316/) (2020)
- [Computing excited states of molecules using normalizing flows](https://mlsanity.com/paper/2308.16468/) (2023)
- [Inverse Design of Quantum Holograms in Three-Dimensional Nonlinear Photonic Crystals](https://mlsanity.com/paper/2102.10344/) (2021)
- [Numerical and geometrical aspects of flow-based variational quantum Monte Carlo](https://mlsanity.com/paper/2203.14824/) (2022)

## Abstract (arXiv)

A central problem in quantum mechanics involves solving the Electronic Schrodinger Equation for a molecule or material. The Variational Monte Carlo approach to this problem approximates a particular variational objective via sampling, and then optimizes this approximated objective over a chosen parameterized family of wavefunctions, known as the ansatz. Recently neural networks have been used as the ansatz, with accompanying success. However, sampling from such wavefunctions has required the use of a Markov Chain Monte Carlo approach, which is inherently inefficient. In this work, we propose a solution to this problem via an ansatz which is cheap to sample from, yet satisfies the requisite quantum mechanical properties. We prove that a normalizing flow using the following two essential ingredients satisfies our requirements: (a) a base distribution which is constructed from Determinantal Point Processes; (b) flow layers which are equivariant to a particular subgroup of the permutation group. We then show how to construct both continuous and discrete normalizing flows which satisfy the requisite equivariance. We further demonstrate the manner in which the non-smooth nature ("cusps") of the wavefunction may be captured, and how the framework may be generalized to provide induction across multiple molecules. The resulting theoretical framework entails an efficient approach to solving the Electronic Schrodinger Equation.